Constitutive relations in dense granular flows

We use simulations to investigate constitutive relations in dry granular flow. Our system is comprised of polydisperse sets of spherical grains falling down a vertical chute under the influence of gravity. Three phases or states of granular matter are observed: a free-fall dilute granular gas region at the top of the chute, a granular fluid in the middle and then a glassy region at the bottom. We examine a complete closed set of constitutive relations capable of describing the local stresses, heat flow, and dissipation in the different regions. While the pressure can be reasonably described by hard sphere gas models, the transport coefficients cannot. Transport coefficients such as viscosity and heat conductivity increase with decreasing temperature in the fluid and glassy phases. The glass exhibits signs of a finite yield stress and we show that the static sand pile is a limit of our glassy state.

Figure 1

(Color online) (a) Section of a simulation involving $43200$ grains with 15% polydispersity. The system size is $32a\times 32a\times 400a$. There are reflective walls at $x=0$ and $x=L_x$, periodic boundary conditions in the $z$ direction, and a finite probability of reflection at the bottom of the chute (at $y=0$) with an asymptotic coefficient of restitution ${\mu }_0=0.97$. Time-averaged density (volume fraction) in (b) and $y-$ velocity in (c) down the center of a 3D chute. The short dashed lines are analytic calculations of density and $v_y$ in the free-fall region (described in text). (d) The total kinetic energy $E=\frac{1}{2}\rho v^2+\frac{3}{2}\rho T$ and (e) stress tensor components ${\sigma }_{xx}$ (solid line), ${\sigma }_{yy}$ (dashed line) and ${\sigma }_{zz}$ (dotted line). The measurements for plots (b), (c), (d), and (e) were taken down the center of the chute. Units are described in footnote [30]. For the color online version, the balls in the 3D visualization are colored according to the magnitude of vertical $y$ velocity with the red balls being the fastest and the blue balls the slowest.

Figure 2

(Color online) Plot of force densities ${\partial }_y{\sigma }_{yy}$ (dot-dashed line), ${\partial }_x{\sigma }_{yx}$ (long dashed line), ${\partial }_x{\sigma }_{yx}+{\partial }_y{\sigma }_{yy}$ (short dashed line) and the weight $-\rho g_y$ (solid line) in (a) the fluid region and (b) the glassy region versus the width $x$ for a 400-height column using an asymptotic coefficient of restitution ${\mu }_0$ of 0.97 and probability of reflection $p=90\mathrm{\%}$. (c) Plot of force densities ${\partial }_y{\sigma }_{yy}$ (dot-dashed line), $-{\partial }_y\rho v_y^2$ (long dashed line), (${\partial }_x{\sigma }_{yy}-{\partial }_y\rho v_y^2$ (short dashed line) and the weight $-\rho g_y$ (solid line) in the free-fall region. Units are described in footnote [30].

Figure 3

(Color online) (a) The components of the energy equation from Eq. (14) in the free-fall region. The solid line is the left side of the equation, $\nabla \cdot \mathbf{F}$, and the dashed line is the right side of the equation, $I+\rho g{v}_y$. Units are described in footnote [30].

Figure 4

(Color online) (a) Plot of $\rho \delta v_x^2/P_{xx}$ (squares), $\rho \delta v_y^2/P_{yy}$ (triangles), and $\rho \delta v_z^2/P_{zz}$ (circles) versus $\varphi$. Also plotted are the result from Eq. (19) [solid green (bottom) line], Eq. (18) [blue dotted (middle) line], and Eq. (23) [orange dot-dashed (top) curve]. ${\varphi }_c$ is the observed close-packed density in the glassy region and $\delta v_{\alpha }^2=⟨{\left(v_{\alpha }-⟨v_{\alpha }⟩\right)}^2⟩$. (b) Inverse of data in (a). In both plots closed symbols indicate the glassy region and open symbols the fluid regions. The data are for a $32\times 32\times 400$ simulation with an asymptotic coefficient of restitution ${\mu }_0=0.97$ and a probability of reflection at the bottom of the chute $p=0.9$. The red curve through the data are described in the text.

Figure 5

(Color online) Plot of the empirically determined parameter $A$ in Eq. (30) for the glassy region of the chute from simulations with different asymptotic coefficients of restitution ${\mu }_0$. The line is just a guide for the eyes.

Figure 6

(Color online) Plot of the collision frequency per unit volume as calculated using Eq. (31) using $A=1.62$ (red line with ○’s), as calculated using Eq. (31) using $A=2$ (purple line with ◻’s), and the simulation values for the collision frequency (blue solid line) versus the height $y$ of the chute in (a) the entire chute and (b) in the fluid region (semilogarithmic). Measurements are taken in the center of the chute using a simulation with an asymptotic coefficient of restitution ${\mu }_0=0.97$. Data are averaged over depth $\left(32a\right)$ in $z$ and over 800 time units. Units are described in footnote [30].

Figure 7

(Color online) (a) Shear stress ${\sigma }_{xy}$ versus shear rate ${\partial }_x{v}_y$ in the fluid region for a slow flow. Lines connect data at fixed heights in the fluid region (▲’s at $y=277$, ◼’s at $y=283$, ●‘s at $y=289$ and ◆’s at 295) and each symbol is from a local measurement at a different $x$ (probability of reflection at the bottom sieve of $\text{p}=90\mathrm{\%}$, ${\mu }_0=0.95$ in all cases). (b) Log-log plot of viscosity in the fluid region [slope of each line in (a)]. Symbols indicate different asymptotic coefficients of restitution, ${\mu }_0$ (with ○’s using ${\mu }_0=0.95$ and $p=90\mathrm{\%}$, ◇’s using ${\mu }_0=0.96$ and $p=90\mathrm{\%}$, and ▽’s using ${\mu }_0=0.97$ and $p=90\mathrm{\%}$). Systems with the higher asymptotic coefficients of restitution of ${\mu }_0=0.95$, 0.96 and 0.97 achieve a true fluid region and have a consistent power-law of $-\frac{4}{3}$.) (c) Same data as (b) but scaled by ${\eta }_0$ from Eq. (42) and ploted versus volume fraction $\varphi$. The lines in (c) are from the Enskog prediction, Eq. (43) as explained in the text. Units are described in footnote [30].

Figure 8

(Color online) (a) Log-Log plot of square root of temperature versus shear rate in the shear zones along the walls of the glassy region. A system with a fast flow (probability of reflection at the bottom sieve of $p=1\mathrm{\%}$, △ and ${\mu }_0=0.9$) yields a region with the experimental power-law exponent of 0.4 (line), while slow systems (probability of reflection at the bottom sieve of $p=90\mathrm{\%}$ with ${\mu }_0=0.9$ ◻, ${\mu }_0=0.95$ ○, ${\mu }_0=0.96$ ◇, and ${\mu }_0=0.97$ ▽) have a pseudo-power-law with exponent 0.2. (b) Log-log plot of effective shear viscosity $\eta ={\sigma }_{xy}/{\partial }_x{v}_y$ versus temperature $T$ in the glassy region for a fast flow (△) and slower flows [symbols same as in (a)]. The solid lines have slopes of $-1.1$ for the fast system, $-2.3$ for the slow system with ${\mu }_0=0.9$, and $-2.8$ for the slow systems with ${\mu }_0=0.95$, 0.96 and 0.97. Units are described in footnote [30].

Figure 9

(Color online) (a) Log-log plot of local shear stress ${\sigma }_{yx}$ versus local shear rate ${\partial }_x{v}_y$ at a constant height in the glassy region for various flow rates (probability of reflection at the bottom sieve of $p=0.01$ as △’s, $p=0.1$ as ◻’s, $p=0.25$ as ○’s, $p=0.5$ as △’s, $p=0.75$ as ▽’s, and $p=0.9$ as ▲’s), all with an asymptotic coefficient of restitution of ${\mu }_0=0.9$. Lines connect data at fixed heights and each symbol is from a local measurement at a different $x$. The solid straight lines have slopes of 0.41 for the upper line and 0.38 for the lower line. (b) Plot of shear stress ${\sigma }_{yx}$ at the wall versus the $y$ velocity at the center at $L_x/2$ minus the $y$ velocity at the wall, scaled by $L_x/2$. The yield stress ${\sigma }_Y$ is indicated by the arrow. Data are from a $32\times 32\times 250$ column. Units are described in footnote [30].

Figure 10

(Color online) (a) Heat flux $Q_x$ versus granular temperature gradient, $-{\partial }_{x}T$, in the fluid region (probability of reflection at the bottom sieve of $\text{p}=90\mathrm{\%}$, ${\mu }_0=0.97$). Data points with the same symbol shape were measured at the same height but different $x$ in the fluid regions (▲’s at $y=274$, ◼’s at $y=284$, ●‘s at $y=294$ and ◆’s at $y=304$). (b) Semilogarithmic plot of thermal conductivity, $\kappa =-Q_x/{\partial }_{x}T$ versus the granular temperature, $T$, in the fluid region for systems with a probability of reflection at the bottom sieve of $p=90\mathrm{\%}$. The symbols indicate different asymptotic coefficients of restitution, ${\mu }_0$ (with ◻’s using ${\mu }_0=0.95$, ○’s using ${\mu }_0=0.96$, and ▽’s using ${\mu }_0=0.97$). (c) Same data as in (b) but scaled by ${\kappa }_0$ [Eq. (57)] and plotted versus volume fraction $\varphi$. The lines are from an Enskog prediction described in the text. Units are described in footnote [30].

Figure 11

(Color online) Log-log plot of heat flux $Q_x=-\kappa {\partial }_{x}3T$ vs width $x$ for a 15% polydisperse 3D simulation for a glassy region at $y=200$ where $\kappa =4/3\pi a^3\rho f_c$. The solid line is $Q_x=F_{cx}+{\sigma }_{xy}{v}_y$, and the dashed line is $Q_x=-4/3\pi ⟨a^3⟩\rho f_c{\partial }_{x}3T$. Units are described in footnote [30].

Figure 12

(Color online) Semilogarithmic plot of the negative of the dissipation, $-I$, from the simulation as calculated from Eq. (11) (blue solid line), as calculated using Eq. (63) (red dashed line), and as calculated using Eq. (64) (purple dotted line). Units are described in footnote [30].

Figure 13

(Color online) Shear stress ${\sigma }_{xy}$ (solid line) and $R_{xy}$ (circles) [right hand side of Eq. (66)], the shear stress factorized into the collision directions, $⟨\left(\hat{\mathbf{q}}\cdot \hat{\mathbf{x}}\right)\left(\hat{\mathbf{q}}\cdot \hat{\mathbf{y}}\right)⟩$ and the constant $-\frac{1}{2}{f}_c⟨\left(1+\mu \right)\left({\dot{\mathbf{r}}}_{\mathbf{1}}-{\dot{\mathbf{r}}}_{\mathbf{2}}\right)\cdot \hat{\mathbf{q}}⟩$) versus $x$ in (a) the glassy region at a height, $y=100$. (b) Plot of the diagonal stress, ${\sigma }_{\alpha \alpha }$ with its kinetic term (lower curves) and without its kinetic term (upper curves), and factor $R_{\alpha \alpha }$ versus height $y$. (${\sigma }_{xx}$ is the solid line, ${\sigma }_{yy}$ is the dashed line, and ${\sigma }_{zz}$ is the dot-dashed line, $R_{xx}$ is circles, $R_{yy}$ is squares, and $R_{zz}$ is triangles). Data are for a 400-height column using an asymptotic coefficient of restitution ${\mu }_0$ of 0.97 and a probability of reflection, $p=0.9$. Units are described in footnote [30].

Figure 14

(Color online) (a) Plot of $f_c⟨\left(1+\mu \right)\left({\dot{\mathbf{r}}}_{\mathbf{1}}-{\dot{\mathbf{r}}}_{\mathbf{2}}\right)\cdot \hat{\mathbf{q}}⟩$ vs height $y$ for a 400-height column. The lines from bottom to top represent data with asymptotic coefficients of restitution ${\mu }_0$ of 0.95, 0.96, and 0.97, all with a sieve reflection probability, $p=0.9$. Data are averaged over the width ($x$ direction). (b) Plot of $f_c⟨\left(1+\mu \right)\left({\dot{\mathbf{r}}}_{\mathbf{1}}-{\dot{\mathbf{r}}}_{\mathbf{2}}\right)\cdot \hat{\mathbf{q}}⟩$ vs height $y$ for a 250-height column. The lines from top to bottom represent data with sieve reflection probabilities $p$ of 0.25, 0.5, and 0.75, all with an asymptotic coefficient of restitution ${\mu }_0=0.9$. Units are described in footnote [30].

Figure 15

(Color online) Plot of the eigenvalues (compressive stresses) and corresponding directions of the stress tensor along width of column in (top) the glassy region at $y=138$, and (bottom) in the fluid region at $y=294$ for a 400-height column using an asymptotic coefficient of restitution ${\mu }_0$ of 0.97. In both (top) and (bottom), the eigenvalues are associated alongside with the eigenvector directions by the style of the lines. That is, the line style of the eigenvalues (shown as solid, dashed, or dotted lines) are matched with the line style of the box (shown as a solid, dashed, or dotted lined box) surrounding the particular eigenvector directions. The analytical solution for the eigenvalues given by Eq. (74) with $c=0$ is plotted as thick lines with ${\Lambda }_1$ drawn in pink, ${\Lambda }_2$ in green and ${\Lambda }_3$ in yellow. Units are described in footnote [30].

Figure 16

(Color online) Shear stress ${\sigma }_{xy}$ scaled by pressure ${\sigma }_{xx}$ raised to the $1-\alpha /2$, with $\alpha =0.39$ for all of the data plotted in Fig. 9a. Again, each symbol is at a different $x$ along a cross section going from the wall into the center of the chute and lines connect data at a common height (and pressure).

Figure 17

(Color online) Inverse of the volume fraction $\varphi$ in the glass (height of $y=125$) versus the strain rate ${\partial }_x{v}_y$ (each symbol is at a different $x$ along a cross section going from the wall into center of the chute) using an asymptotic coefficient of restitution ${\mu }_0$ of 0.97. Units are described in footnote [30].